The sharp lifespan of small data smooth solutions to 2-D quadratic quasilinear wave equations in exterior domains

Abstract

In the paper [M. Keel, H. Smith, C.D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions. J. Amer. Math. Soc. 17 (2004), no. 1, 109-153], the authors prove that for the 3-D quadratic quasilinear wave equation in exterior domains with homogenous Dirichlet boundary value and small initial data of size $\varepsilon$, the lifespan ${\bar T}_{\varepsilon}$ of the smooth solution fulfills ${\bar T}_{\varepsilon}\ge e^{C/\varepsilon}$. However, for the corresponding 2-D quadratic quasilinear wave equation in exterior domains with homogenous Dirichlet or Neumann boundary value, so far it is still open whether the expected sharp lifespan $T_{\varepsilon}\ge\frac{C}{\varepsilon^2}$ holds or not. In this paper, we will solve this open question. Our main ingredients include: introducing the suitable Friedlander radiation field for the 2-D linear wave equation in exterior domains with homogenous Dirichlet or Neumann boundary value, constructing the delicate approximate solution, and establishing some crucial space-time decay estimates for the solutions of 2-D quasilinear wave equation in exterior domains. On the other hand, for the radial symmetric solutions to a class of 2-D quadratic quasilinear wave equation in exterior domains, the upper bound of the lifespan $T_{\varepsilon}\le\frac{C}{\varepsilon^2}$ is derived and the sharp constant $C$ is also determined explicitly.